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Cited by 76Financial Markets and Investment StrategiesFinancial Risk and Volatility ModelingRisk and Portfolio Optimization

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Entropy-Based Financial Asset Pricing

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Ormos, Mihály, Zibriczky, Dávid·2014·PLoS ONE
DOI 10.1371/journal.pone.0115742PMC PMC4278763
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Abstract

We investigate entropy as a financial risk measure. Entropy explains the equity premium of securities and portfolios in a simpler way and, at the same time, with higher explanatory power than the beta parameter of the capital asset pricing model. For asset pricing we define the continuous entropy as an alternative measure of risk. Our results show that entropy decreases in the function of the number of securities involved in a portfolio in a similar way to the standard deviation, and that efficient portfolios are situated on a hyperbola in the expected return-entropy system. For empirical investigation we use daily returns of 150 randomly selected securities for a period of 27 years. Our regression results show that entropy has a higher explanatory power for the expected return than the capital asset pricing model beta. Furthermore we show the time varying behavior of the beta along with entropy.

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Figure 1 Average value of risk and risk reduction vs. number of securities in portfolio. Note: We generate 10 million random equally weighted portfolios with different number of securities involved (at most 100,000 for each size) based on 150 randomly selected securities from S&P500. The risk of portfolios is estimated by standard deviation (gray continuous curve), Shannon- (black continuous curve) and Rényi entropy (black dashed curve) in the period from 1985 to the end of 2011. Both types of entropy functions are calculated by histogram based density function estimation. The left chart shows the average risk estimates for each portfolio size, and the right chart shows the risk reduction compared to an average risk of single security portfolio.
Figure 1 Average value of risk and risk reduction vs. number of securities in portfolio. Note: We generate 10 million random equally weighted portfolios with different number of securities involved (at most 100,000 for each size) based on 150 randomly selected securities from S&P500. The risk of portfolios is estimated by standard deviation (gray continuous curve), Shannon- (black continuous curve) and Rényi entropy (black dashed curve) in the period from 1985 to the end of 2011. Both types of entropy functions are calculated by histogram based density function estimation. The left chart shows the average risk estimates for each portfolio size, and the right chart shows the risk reduction compared to an average risk of single security portfolio.CC BY · PubMed Central
Figure 2 Portfolios with different number of securities involved in E(r) – risk system. Note: The panels show the expected risk premium of the portfolios (calculated by the average of daily risk premiums) versus the estimated risk using different methods; the number of securities involved is indicated by the different markers. We generate a sample of 750 random portfolios by using 150 randomly selected securities and 200-200 random equally weighted portfolios with 2, 5 and 10 securities. The risk of portfolios is estimated by standard deviation, CAPM beta, Shannon- and Rényi entropy by using daily returns in the period from 1985 to the end of 2011. Both types of entropy functions are calculated by histogram based density function estimation, with 175 bins for Shannon entropy and 50 bins for Rényi entropy.
Figure 2 Portfolios with different number of securities involved in E(r) – risk system. Note: The panels show the expected risk premium of the portfolios (calculated by the average of daily risk premiums) versus the estimated risk using different methods; the number of securities involved is indicated by the different markers. We generate a sample of 750 random portfolios by using 150 randomly selected securities and 200-200 random equally weighted portfolios with 2, 5 and 10 securities. The risk of portfolios is estimated by standard deviation, CAPM beta, Shannon- and Rényi entropy by using daily returns in the period from 1985 to the end of 2011. Both types of entropy functions are calculated by histogram based density function estimation, with 175 bins for Shannon entropy and 50 bins for Rényi entropy.CC BY · PubMed Central
Figure 3 Explanatory power of risk measures in long term. Note: The four panels show the relationship between risk premium and risk (standard deviation, CAPM beta, Shannon- and Rényi entropy) of 150 randomly selected securities by using different estimation methods. Both types of entropy functions are calculated by histogram based density function estimation, with 175 bins for Shannon entropy and 50 bins for Rényi entropy. The equation and the explanatory power (R 2) of the linear regressions are presented using expected risk premium as target variable and risk as explanatory variable. Under the OLS regression equations in brackets the p-values can be seen for each parameter estimations. The R2s of the models applying entropy based risk measures are significantly different form standard deviation and CAPM beta at 1% level.
Figure 3 Explanatory power of risk measures in long term. Note: The four panels show the relationship between risk premium and risk (standard deviation, CAPM beta, Shannon- and Rényi entropy) of 150 randomly selected securities by using different estimation methods. Both types of entropy functions are calculated by histogram based density function estimation, with 175 bins for Shannon entropy and 50 bins for Rényi entropy. The equation and the explanatory power (R 2) of the linear regressions are presented using expected risk premium as target variable and risk as explanatory variable. Under the OLS regression equations in brackets the p-values can be seen for each parameter estimations. The R2s of the models applying entropy based risk measures are significantly different form standard deviation and CAPM beta at 1% level.CC BY · PubMed Central
Figure 4 Explanatory power of risk measures in long term by diversification. Note: This figure shows the explanatory power (R 2) of portfolios with different number of securities involved and different risk estimate methods. We generate 10 million random equally weighted portfolios with different number of securities involved (at most 100,000 for each size) using daily risk premiums of 150 randomly selected securities. The risk of portfolios is estimated by standard deviation (light gray curve), CAPM beta (black dotted curve), Shannon- (gray) and Rényi entropy (black). Both types of entropy functions are calculated by histogram based density function estimation, with 175 bins for Shannon entropy and 50 bins for Rényi entropy.
Figure 4 Explanatory power of risk measures in long term by diversification. Note: This figure shows the explanatory power (R 2) of portfolios with different number of securities involved and different risk estimate methods. We generate 10 million random equally weighted portfolios with different number of securities involved (at most 100,000 for each size) using daily risk premiums of 150 randomly selected securities. The risk of portfolios is estimated by standard deviation (light gray curve), CAPM beta (black dotted curve), Shannon- (gray) and Rényi entropy (black). Both types of entropy functions are calculated by histogram based density function estimation, with 175 bins for Shannon entropy and 50 bins for Rényi entropy.CC BY · PubMed Central
Figure 5 Explanatory power of risk measures in bullish sample. Note: The panels show the relationship between the expected risk premium of securities and risk by using different estimation methods. We present the equation of linear regression and the goodness of fit (R 2).We estimated the risk of 150 random securities in upward trend periods (bull market) from 1985 to the end of 2011 using standard deviation, CAPM beta, Shannon- and Rényi entropy risk estimation methods. Both types of entropy functions are calculated by histogram based density function estimation, with 175 bins for Shannon entropy and 50 bins for Rényi entropy. Under the OLS regression equations in brackets the p-values can be seen for each parameter estimations. The R2s of the models applying entropy based risk measures are significantly higher than the models with standard deviation and CAPM beta at 1% level.
Figure 5 Explanatory power of risk measures in bullish sample. Note: The panels show the relationship between the expected risk premium of securities and risk by using different estimation methods. We present the equation of linear regression and the goodness of fit (R 2).We estimated the risk of 150 random securities in upward trend periods (bull market) from 1985 to the end of 2011 using standard deviation, CAPM beta, Shannon- and Rényi entropy risk estimation methods. Both types of entropy functions are calculated by histogram based density function estimation, with 175 bins for Shannon entropy and 50 bins for Rényi entropy. Under the OLS regression equations in brackets the p-values can be seen for each parameter estimations. The R2s of the models applying entropy based risk measures are significantly higher than the models with standard deviation and CAPM beta at 1% level.CC BY · PubMed Central
Figure 6 Explanatory power of risk measures in bearish sample. Note: The panels show the relationship between the expected risk premium of securities and risk by using different estimation methods. We present the equation of linear regression and the goodness of fit (R 2).We estimated the risk of 150 random securities in downward trend periods (bear market) from 1985 to the end of 2011 using standard deviation, CAPM beta, Shannon- and Rényi entropy risk estimation methods. Both types of entropy functions are calculated by histogram based density function estimation, with 175 bins for Shannon entropy and 50 bins for Rényi entropy. Under the OLS regression equations in brackets the p-values can be seen for each parameter estimations. The R 2 of the models applying entropy based risk measures are significantly higher than the model with standard deviation at 1% level.
Figure 6 Explanatory power of risk measures in bearish sample. Note: The panels show the relationship between the expected risk premium of securities and risk by using different estimation methods. We present the equation of linear regression and the goodness of fit (R 2).We estimated the risk of 150 random securities in downward trend periods (bear market) from 1985 to the end of 2011 using standard deviation, CAPM beta, Shannon- and Rényi entropy risk estimation methods. Both types of entropy functions are calculated by histogram based density function estimation, with 175 bins for Shannon entropy and 50 bins for Rényi entropy. Under the OLS regression equations in brackets the p-values can be seen for each parameter estimations. The R 2 of the models applying entropy based risk measures are significantly higher than the model with standard deviation at 1% level.CC BY · PubMed Central

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  1. Turlach BA (1993) Bandwidth selection in kernel density estimation: A review: Université catholique de Louvain. DOI: 10.1.1.44.6770.

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Metadata

Authors
Ormos, M., & Zibriczky, D.
Affiliations
Budapest University of Technology and Economics (HU)
Corresponding author
Ormos, Mihály
ORCID
Ormos, Mihály
Venue
PLoS ONE
Publisher
Public Library of Science (PLoS)
Published
2014-12-29
Volume
9
Issue
12
Pages
e115742
ISSN
1932-6203 (ISSN-L)
DOI
10.1371/journal.pone.0115742
PMC
PMC4278763
OpenAlex
W1964018898
Cited by
70
References
24
Citations by year
2026: 2 · 2025: 6 · 2024: 1 · 2023: 1 · 2022: 6 · 2021: 7 · 2020: 17 · 2019: 12 · 2018: 4 · 2017: 9 · 2016: 3 · 2015: 2
FWCI
8.54
Type
journal article
Language
en
Open access
Licence
Locations
  • PLoS ONE (publishedVersion, cc-by)
  • Europe PMC (PubMed Central) (publishedVersion, cc-by)
  • PubMed Central (publishedVersion, cc-by)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • RePEc: Research Papers in Economics (submittedVersion)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
  • Figshare (submittedVersion, cc-by)
Topics
  • Social Sciences › Economics, Econometrics and Finance › Finance › Financial Markets and Investment Strategies
  • Social Sciences › Economics, Econometrics and Finance › Finance › Financial Risk and Volatility Modeling
  • Social Sciences › Decision Sciences › Management Science and Operations Research › Risk and Portfolio Optimization
Keywords
entropy (information theory), asset pricing, equity premium, capital asset pricing model, beta, efficient frontier, expected returns, portfolio diversification, time-varying beta, daily stock returns
MeSH terms
Costs and Cost Analysis, Investments, Models, Theoretical, Risk Factors, Entropy

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